Question

Six machines, each working at the same constant rate, together can complete a certain job in 12 days. How many additional machines, each working at the same constant rate, will be needed to complete the job in 8 days?

Answer 1

$six \: machines \: can \: do \\ \: work \: in \: 12 \: days \\ \frac{6}{x} = \frac{1}{12} \\ x = 72 \\ now\: each \: machine \: can \: do \: in \: 72 \: days \: \\ let \: x \: machines \: are \: needed \\ \: to \: work \: in \: 8 \: days \\ then \\ \frac{x}{72} = \frac{1}{8} \\ x = 9 \\ hence \: 3 \: more \: machine \: will \:be \: needed$

Answer 2

The number of days needed is inversely proportion to the number of machines needed.

⇒ The more days we have, the fewer machines we need

⇒ The fewer days we have, the more machines we need

16 machines can finish the job in 12 days:

⇒ 12 days = 6 machines

⇒ 1 day = 12 x 6 = 72 machines

Find the number of machines needed if we have 8 days:

⇒ 1 day = 72 machines

⇒ 8 days = 72 ÷ 8 = 9 machines

Number of additional machines needed:

Addition machines needed = 9 - 6 = 3

Answer: 3 additional machines are needed to finish the work in 8 days